We investigate quantum correlations in the ground state of noninteracting Fermi gases of N particles trapped by an external space-dependent harmonic potential, in any dimension. For this purpose, we compute one-particle correlations, particle fluctuations and bipartite entanglement entropies of extended space regions, and study their large-N scaling behaviors. The half-space von Neumann entanglement entropy is computed for any dimension, obtaining S_HS = c_l N^(d-1)/d ln N, analogously to homogenous systems, with c_l=1/6, 1/(6\sqrt{2}), 1/(6\sqrt{6}) in one, two and three dimensions respectively. We show that the asymptotic large-N relation S_A\approx π^2 V_A/3, between the von Neumann entanglement entropy S_A and particle variance V_A of an extended space region A, holds for any subsystem A and in any dimension, analogously to homogeneous noninteracting Fermi gases.
We study some aspects of equilibrium and off equilibrium quantum dynamics of dilute bosonic gases in the presence of a trapping potential. We consider systems with a fixed number of particles N and study their scaling behavior with increasing the trap size. We focus on one-dimensional (1D) bosonic systems, such as gases described by the Lieb-Liniger model and its Tonks-Girardeau limit of impenetrable bosons, and gases constrained in optical lattices as described by the Bose-Hubbard model. We study their quantum (zero-temperature) behavior at equilibrium and off equilibrium during the unitary t…
From the unified statistical thermodynamics of quantum gases, the virial coefficients of ideal Bose and Fermi gases which are trapped under generic power law potential are derived systematically. From the general result of virial coefficients, one can produce the known results in $d=3$ and $d=2$. But more importantly we found that, the virial coefficients of bosons and fermions become equal (except the the second virial coefficient, where the sign is different) when we trap the gases under harmonic potential in $d=1$. This result suggests the equivalence between Bose and Fermi gases which is a…
We investigate the critical properties of cold bosonic gases in three dimensions, confined by an external quadratic potential coupled to the particle density, and realistically described by the Bose-Hubbard (BH) model. The trapping potential is often included in experiments with cold atoms and modifies the critical finite-size scaling of the homogeneous system in a non trivial way. The trap-size scaling (TSS) theory accounts for this effect through the exponent $θ$. We perform extensive simulations of the BH model at the critical temperature, in the presence of harmonic traps. We find that the…
We investigate the particle-number dependence of some features of the out-of-equilibrium dynamics of d-dimensional Fermi gases in the dilute regime. We consider protocols entailing the variation of the external potential which confines the particles within a limited spatial region, in particular sudden changes of the trap size. In order to characterize the dynamic behavior of the Fermi gas, we consider various global quantities such as the ground-state fidelity for different trap sizes, the quantum work statistics associated with the protocol considered, and the Loschmidt echo measuring the ov…
We theoretically examine equilibrium properties of the harmonically trapped ideal Bose and Fermi gases in the quantum degeneracy regime. We analyze thermodynamic characteristics of gases with a finite number of atoms by means of the known semiclassical approach and perform comparison with exact numerical results. For a Fermi gas, we demonstrate deviations in the Fermi energy values originating from a discrete level structure and show that these are observable only for a small number of particles. For a Bose gas, we observe characteristic softening of phase transition features, which contrasts …
The problem of particle fluctuations in arbitrary nonuniform systems with Bose-Einstein condensate is considered. This includes the case of trapped Bose atoms. It is shown that the correct description of particle fluctuations for any nonuniform system of interacting atoms always results in thermodynamically normal fluctuations.
We study population imbalanced Fermi mixtures under quasi-two-dimensional confinement at zero temperature. Using mean-field theory and the local-density approximation, we study the ground state configuration throughout the BEC-BCS crossover. We find the trapped system to be either fully normal or to consist of a superfluid core surrounded by a normal shell, which is itself either fully or partially polarized. Upon changing the trap imbalance, the trap configuration may undergo continuous transitions between the different ground states. Finally, we argue that thermal equilibration throughout th…
A unified description for the Bose and Fermi gases trapped in an external generic power law potential $U=\sum_{i=1} ^d c_i |\frac{x_i}{a_i}|^{n_i}$ is presented using the grandpotential of the system in $d$ dimensional space. The thermodynamic quantities of the quantum gases are derived from the grand potential. An equivalence between the trapped Bose and Fermi gases is constructed in one dimension ($d=1$) using the Landen relation. It is also found that the established equivalence between the ideal free Bose and Fermi gases in $d=2$ (M. H. Lee, Phys. Rev. E 55, 1518 (1997)) is lost when exter…
Yoram Alhassid conjectured that the total energy of a harmonically trapped two-component Fermi gas with large scattering length is a linear functional of the occupation probabilities of single-particle energy eigenstates. We confirm his conjecture and derive the functional explicitly. We show that the functional applies to ALL smooth potentials having a minimum, not just harmonic traps. We also calculate the occupation probabilities of high energy states.
We investigate thermodynamics and quantum criticality of strongly attractive Fermi gases confined in a one-dimensional (1D) harmonic trap. Finite temperature density profiles, entropy, compressibility and susceptibility of the trapped gas are studied using analytic results for the thermodynamics within the local density approximation. We demonstrate that current experiments are capable of measuring universal Tomonaga-Luttinger liquid physics and quantum criticality of 1D strongly interacting Fermi gases. The results provide insights on recent measurements of key features of the phase diagram o…
We demonstrate a novel class of trapping potentials, time-averaged adiabatic potentials (TAAP) which allows the generation of a large variety of traps and waveguides for ultracold atoms. Multiple traps can be coupled through controllable tunneling barriers or merged altogether. We present analytical expressions for pancake-, cigar-, and ring- shaped traps. The ring-geometry is of particular interest for guided matter-wave interferometry as it provides a perfectly smooth waveguide of controllable diameter, and thus a tunable sensitivity of the interferometer.
The position and momentum space information entropies of weakly interacting trapped atomic Bose-Einstein condensates and spin-polarized trapped atomic Fermi gases at absolute zero temperature are evaluated. We find that sum of the position and momentum space information entropies of these quantum systems contain $N$ atoms confined in a $D(\le 3)$-dimensional harmonic trap has a universal form as $ S_t^{(D)} = N(a D - b \ln N) $, where $a \simeq 2.332$ and $b = 2$ for interacting bosonic systems and $a \simeq 1.982$ and $b = 1$ for ideal fermionic systems. These results obey the entropic uncert…
A two-component Bose-Einstein condensate that is confined in a one-dimensional ring potential supports solitary-wave solutions, which we evaluate analytically. The derived solutions are shown to be unique. The corresponding dispersion relation that generalizes the case of a single-component system shows interesting features.
We report on the confinement of a Bose-Einstein condensate in an annular trap with widely tunable parameters. The trap relies on a combination of magnetic, optical and radio-frequency fields. The loading procedure is discussed. We present annular traps with radii adjusted between 20 and 150 micrometers. We demonstrate the preparation of persistent flows both with a rotating laser stirrer and with a global quadrupole deformation of the ring.Our setup is well adapted for the study of superfluid dynamics.
A two-component Bose-Einstein condensate confined in an axially-symmetric potential with two local minima, resembling two concentric annular traps, is investigated. The system shows a number of quantum phase transitions that result from the competition between phase coexistence, and radial/azimuthal phase separation. The ground-state phase diagram, as well as the rotational properties, including the (meta)stability of currents in this system, are analysed.
We study the virial relations for ultracold trapped two component Fermi gases in the case of short finite range interactions. Numerical verifications for such relations are reported through the BCS-BEC crossover. As an intermediate step, it is necessary to evaluate the partial derivatives of the many body energy with respect to the inverse of the scattering length and with respect to the interaction range. They are found to have extreme values at the unitary limit. The virial results are used to check the quality of the variational wave function involved in the calculations.
For two-dimensional (2D) atomic Fermi gases in harmonic traps, the SO(2,1) symmetry is broken by the interatomic interaction explicitly via the contact correlation operator. Consequently the frequency of the breathing mode $ω_B$ of the 2D Fermi gas can be different from $2ω_0$, with $ω_0$ the trapping frequency of harmonic potentials. At zero temperature, we use the sum rules of density correlation functions to yield upper bounds for $ω_B$. We further calculate $ω_B$ through the Euler equations in the hydrodynamic regime. The obtained value of $ω_B$ satisfies the upper bounds and shows deviati…
We study density profiles in trapped fermionic gases, near Feshbach resonances, at all $T \leq T_c$ and in the near-BEC and unitary regimes. For the latter, we quantify and characterize the generally neglected contribution from noncondensed Cooper pairs. As a consequence of these pairs, our profiles are rather well fit to a Thomas-Fermi (TF) functional form, and equally well fit to experimental data. Our work lends support to the notion that TF fits can be used in an experimental context to obtain information about the temperature.
It is quite common that several different phases exist simultaneously in a system of trapped quantum gases of ultra-cold atoms. One such example is the strongly-interacting Fermi gas with two imbalanced spin species, which has received a great amount of attention due to the possible presence of exotic superfluid phases. By employing novel numerical techniques and algorithms, we self-consistently solve the Bogoliubov de-Gennes equations, which describe Fermi superfluids in the mean-field framework. From this study, we investigate the novel phases of spin-imbalanced Fermi gases and examine the v…
The two-component Fermi gas in a double-well trap is studied using the density functional theory and the density profile of each component is calculated within the Thomas-Fermi approximation. We show that the two components are spatially separate in the two wells once the repulsive interaction exceeds the Stoner point, signaling the occurrence of the ferromagnetic transition. Therefore, the double-well trap helps to explore itinerant ferromagnetism in atomic Fermi gases, since the spontaneous separation can be examined by measuring component populations in one well.
The highly controllable ultracold atoms in a one-dimensional (1D) trap provide a new platform for the ultimate simulation of quantum magnetism. In this regard, the Neel-antiferromagnetism and the itinerant ferromagnetism are of central importance and great interest. Here we show that these magnetic orders can be achieved in the strongly interacting spin-1/2 trapped Fermi gases with additional p-wave interactions. In this strong coupling limit, the 1D trapped Fermi gas exhibit an effective Heisenberg spin XXZ chain in the anisotropic p-wave scattering channels. For a particular p-wave attractio…
We examine the problem of stability of persistent currents in a mixture of two Bose gases trapped in an annular potential. We evaluate the critical coupling for metastability in the transition from quasi-one to two-dimensional motion. We also evaluate the critical coupling for metastability in a mixture of two species as function of the population imbalance. The stability of the currents is shown to be sensitive to the deviation from one-dimensional motion.
I study vortex ring oscillations in a superfluid, trapped in an elongated trap, under the conditions of the Local Density Approximation. On the basis of the Hamiltonian formalism I develop a hydrodynamic theory, which is valid for an arbitrary superfluid and depends only on the equation of state. The problem is reduced to an ordinary differential equation for the ring radius. The cases of the dilute BEC and the Fermi gas at unitarity are investigated in detail. Simple analytical equations for the periods of small oscillations are obtained and the equations of non-linear dynamics are solved in …
Many strongly coupled fluids are known to share similar hydrodynamic transport properties. In this work we argue that this similarity could extend beyond hydrodynamics to transient dynamics through the presence of non-hydrodynamic modes. We review non-hydrodynamic modes in kinetic theory and gauge/gravity duality and discuss their signatures in trapped Fermi gases close to unitarity. Reanalyzing previously published experimental data, we find hints of non-hydrodynamic modes in cold Fermi gases in two and three dimensions.
We measure the temporal pair correlation function $g^{(2)}(τ)$ of a trapped gas of bosons above and below the critical temperature for Bose-Einstein condensation. The measurement is performed {\it in situ} using a local, time-resolved single-atom sensitive probing technique. Third and fourth order correlation functions are also extracted. We develop a theoretical model and compare it with our experimental data, finding good quantitative agreement and highlighting the role of interactions. Our results promote temporal correlations as new observables to study the dynamics of ultracold quantum ga…
We report on trapping and cooling Li-6 atoms in a monolithic ring bowtie cavity. To make the cavity insensitive to magnetic fields used to tune atomic interactions, we constructed it entirely from fused silica and Zerodur. The components were assembled using hydroxide bonding, which we show can be compatible with ultra-high vacuum. Backscattering in high-finesse ring cavities readily causes trap intensity fluctuations and heating, but with phase-controlled bi-directional pumping the trap lifetime can be made long enough for quantum gas experiments in both the crossed-beam trap (unidirectional …
By applying the recently proposed antisymmetric superfluid local density approximation (ASLDA) to strongly interacting polarized atomic gases at unitarity in very elongated traps, we find families of Larkin-Ovchinnikov (LO) type of solutions with prominent transversal oscillation of pairing potential. These LO states coexist with a superfluid state having a smooth pairing potential. We suggest that the LO phase could be accessible experimentally by increasing adiabatically the trap aspect ratio. We show that the local asymmetry effects contained in ASLDA do not support a deformed superfluid co…
We analyze the phase structure of mass- and spin-imbalanced unitary Fermi gases in harmonic traps. To this end, we employ Density Functional Theory in the local density approximation. Depending on the values of the control parameters measuring mass and spin imbalance, we observe that three regions exist in the trap, namely: a superfluid region at the center, surrounded by a mixed region of resonantly interacting spin-up and spin-down fermions, and finally a fully polarized phase surrounding the previous two regions. We also find regimes in the phase diagram where the existence of a superfluid …
Density correlations unambiguously reveal the quantum nature of matter. Here, we study correlations between measurements of density in cold-atom clouds at different times at one position, and also at two separated positions. We take into account the effects of finite-size and -duration measurements made by light beams passing through the atom cloud. We specialise to the case of Bose gases in harmonic traps above critical temperature, for weakly-perturbative measurements. For overlapping measurement regions, shot-noise correlations revive after a trap oscillation period. For non-overlapping reg…
We study the finite temperature $T$ behavior of trapped Fermi gases as they undergo BCS-Bose Einstein condensation (BEC) crossover, in the presence of a population imbalance. Our results, in qualitative agreement with recent experiments, show how the superfluid phase transition is directly reflected in the particle density profiles. We demonstrate that at $ T \neq 0$ and in the near-BEC and unitary regimes, the polarization is excluded from the superfluid core. Nevertheless a substantial polarization fraction is carried by a normal region of the trap having strong pair correlations, which we a…
Within the context of Hartree-Fock-Bogoliubov theory, we study the behavior of superfluid Fermi systems when they pass from a small to a large container. Such systems can be now realized thanks to recent progress in experimental techniques. It will allow to better understand pairing properties at overflow and in general in rapidly varying external potentials.
Novel ground state properties of rotating Bose gases have been intensively studied in the context of neutral cold atoms. We investigate the rotating Bose gas in a trap from a thermodynamic perspective, taking the charged ideal Bose gas in magnetic field (which is equivalent to a neutral gas in a synthetic magnetic field) as an example. It is indicated that the Bose-Einstein condensation temperature is irrelevant to the magnetic field, conflicting with established intuition that the critical temperature decreases with the field increasing. The specific heat and Landau diamagnetization also exhi…
Advances in light shaping for optical trapping of neutral particles have led to the development of box traps for ultracold atoms and molecules. These traps have allowed the creation of homogeneous quantum gases and opened new possibilities for studies of many-body physics. They simplify the interpretation of experimental results, provide more direct connections with theory, and in some cases allow qualitatively new, hitherto impossible experiments. Here we review progress in this emerging field.
We compute the fraction of closed-channel molecules in trapped atomic Fermi gases, over the entire range of accessible fields and temperatures. We use a two-channel model of BCS--Bose-Einstein condensation (BEC) crossover theory at general temperature $T$, and show that this fraction provides a measure of the $T$ dependent pairing gap. Our calculations, containing no free parameters, are in good quantitative agreement with recent low $T$ measurements in $^6$Li. We present readily testable predictions for the dependencies of the closed-channel fraction on temperature and Fermi momentum.
We study the virial expansion for three-dimensional Bose and Fermi gases at finite temperature using an approximation that only considers two-body processes and is valid for high temperatures and low densities. The first virial coefficients are computed and the second is exact. The results are obtained for the full range of values of the scattering length and the unitary limit is recovered as a particular case. A weak coupling expansion is performed and the free case is also obtained as a proper limit. The influence of an anisotropic harmonic trap is considered using the Local Density Approxim…
We address recent momentum resolved radio frequency (RF) experiments on ultracold trapped Fermi gases of $^{40}$K. We show that momentum resolved RF probes provide measurements of the centrally important fermionic spectral function. They also serve to remove ambiguity plaguing the interpretation of momentum integrated RF experiments by establishing a clear dispersion signature of pairing. We find that the temperature dependence of the spectral function is dramatic at unitarity, and, importantly, smooth from above to below $T_c$ throughout BCS-BEC crossover. This should be tested experimentally…
We use the Bogoliubov-de Gennes formalism to analyze harmonically trapped Fermi gases with Rashba-type spin-orbit coupling in two dimensions. We consider both population-balanced and -imbalanced Fermi gases throughout the BCS-BEC evolution, and study the effects of spin-orbit coupling on the spontaneously induced countercirculating mass currents and the associated intrinsic angular momentum. In particular, we find that even a small spin-orbit coupling destabilizes Fulde-Ferrel-Larkin-Ovchinnikov (FFLO)-type spatially modulated superfluid phases as well as the phase-separated states against the…
We present a theoretical interpretation of radio-frequency (RF) pairing gap experiments in trapped atomic Fermi gases, over the entire range of the BCS-BEC crossover, for temperatures above and below $T_c$. Our calculated RF excitation spectra, as well as the density profiles on which they are based, are in semi-quantitative agreement with experiment. We provide a detailed analysis of the physical origin of the two different peak features seen in RF spectra, one associated with nearly free atoms at the edge of the trap, and the other with (quasi-)bound fermion pairs.
We have analytically explored the quantum phenomenon of particle scattering by harmonically trapped Bose and Fermi gases with the short ranged (Fermi-Huang $δ^3_p$ [1]) interactions among the incident particle and the scatterers. We have predicted differential scattering cross-sections and their temperature dependence in this regard. Coherent scattering even by a single boson or fermion in the finite geometry gives rise to new tool of determining energy eigenstate of the scatterer. Our predictions on the differential scattering cross-sections, can be tested within the present day experimental …
The prospect of controlling entanglement in interacting quantum systems offers a myriad of technological and scientific promises, given the progress in experimental studies in systems such as ultracold trapped gases. This control is often challenging because of decoherence, the process by which environmental interactions create spurious entanglements that can destroy the desired entanglement. Considering the collisional decoherence that is relevant for quantum measurements utilizing scattering in one-dimensional trapped gases, here we derive a relationship between particle masses and wave pack…
Trapped ultracold Fermi gases provide a system that can be tuned between the BCS and BEC regimes by means of a magnetic-field Feshbach resonance. Condensation of fermionic atom pairs in a $^{40}$K gas was demonstrated experimentally by a sweep technique that pairwise projects fermionic atoms onto molecules. In this paper we examine previous data obtained with this technique that probed the phase boundary in the temperature-magnetic field plane. Comparison of the $^{40}$K data to a theoretically computed phase diagram demonstrates good agreement between the two.
We present phase diagrams for population imbalanced, trapped Fermi superfluids near unitarity. In addition to providing quantitative values for the superfluid transition temperature, the pairing onset temperature and the transition line (separating the Sarma and phase separation regimes), we study experimental signatures of these transitions based on density profiles and density differences at the center. Predictions on the BCS side of resonance show unexpected behavior, which should be searched for experimentally.
We determine the temperature $T$ dependence of first and second sound mode frequencies for trapped Fermi gases undergoing BCS to Bose Einstein condensation (BEC) crossover. Our results are based on the two fluid equations in conjunction with a microscopic calculation of thermodynamical variables. As in experiment and at unitarity, we show that the lowest radial breathing mode is $T$ independent. At finite $T$, higher order breathing modes strongly mix with second sound. Their complex $T$ dependence should provide an alternative way of measuring the transition temperature, $T_c$.
